Quantum Implications of Non-Extensive Statistics - Cabo Bizet, Nana; Damián, César; Obregón, Octavio; Santos-Silva, Roberto (Frontiers in Physics, 2021)
Exploring the analogy between quantum mechanics and statistical mechanics we formulate an integrated version of the Quantropy functional Baez and Pollard (2015). With this prescription we compute the propagator associated to Boltzmann-Gibbs statistics in the semiclassical approximation as K=F(T)exp(iScl/hbar􏰛). We determine also propagators associated to different non-additive statistics; those are the entropies depending only on the probability S± Obregón (2010) and Tsallis entropy Sq Tsallis (1988). For S± we obtain a power series solution for the probability vs. the energy, which can be analytically continued to the complex plane, and employed to obtain the propagators. Our work is motivated by Nobre et al. (2011) where a modified q-Schrödinger equation is obtained; that provides the wave function for the free particle as a q-exponential. The modified q-propagator obtained with our method, leads to the same q-wave function for that case. The procedure presented in this work allows to calculate q-wave functions in problems with interactions; determining non-linear quantum implications of non-additive statistics. In a similar manner the corresponding generalized wave functions associated to S± can also be constructed. The corrections to the original propagator are explicitly determined in the case of a free particle and the harmonic oscillator for which the semi-classical approximation is exact.